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A model for episodic degassing of an andesitic magma
intrusion
Marie Boichu, Benoıt Villemant, Georges Boudon
To cite this version:
Marie Boichu, Benoıt Villemant, Georges Boudon. A model for episodic degassing of an andesitic magma intrusion. Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2008, �10.1029/2007JB005130�. �insu-01288661�
A model for episodic degassing of an andesitic
magma intrusion
Marie Boichu,1,2 Benoıˆt Villemant,1,3and Georges Boudon1
Received 23 April 2007; revised 9 February 2008; accepted 12 March 2008; published 9 July 2008.
[1] Episodic magmatic degassing has been observed at numerous volcanoes, especially
those of intermediate composition. It can span timescales from years to decades. Here we propose a physical model for the degassing of a shallow magma intrusion to explain this phenomenon. The magma cools by convection, which leads to melt crystallization, volatile exsolution, and magma overpressure. When the pressure reaches a critical value, wall rocks fracture and the exsolved gas escapes. The intrusion then returns to the initial lithostatic pressure and a new cooling-crystallization-degassing cycle occurs. A series of such cycles leads to episodic degassing. The trend and timescale of the degassing process are mainly governed by magma cooling. Two degassing regimes are exhibited: an early phase with a high frequency of gas pulses and a later phase with a lower gas pulse frequency. The transition between these two regimes is caused by the viscosity increase when the magma crystallinity exceeds the crystal percolation threshold. We find that the time to this transition is dependent on magma volume, to a first approximation. Where observations are available from sustained geochemical surveillance, the model provides constraints on key aspects of the subsurface magmatic system, with estimation of the volume of an intrusion and tensile strength of the surrounding rocks. It therefore represents a relevant tool for volcanic surveillance and hazard assessment.
Citation: Boichu, M., B. Villemant, and G. Boudon (2008), A model for episodic degassing of an andesitic magma intrusion, J. Geophys. Res., 113, B07202, doi:10.1029/2007JB005130.
1. Introduction
[2] Noneruptive episodic degassing is a common feature
at many intermediate to silicic volcanoes worldwide. Such degassing is typically sustained for years to decades, with nonrandom, short-duration gas ‘‘crises’’ superimposed on a longer-term secular degassing trend. Volcanoes where such behavior has been observed include La Soufrie`re de Gua-deloupe (Lesser Antilles, France) [Villemant et al., 2005], Campi Flegrei (Italy) [Chiodini et al., 2003], Galeras (Colombia) [Fischer et al., 1996], Vulcano Island (Italy) [Nuccio and Paonita, 2001], Poa´s (Costa Rica) [Rowe et al., 1992], White Island (New Zealand) [Giggenbach and Sheppard, 1989] (Figure 1). The mechanism of magma convection in a conduit has been proposed to explain long-term regular and intense degassing of volatiles from andesitic magma chambers [Stevenson and Blake, 1998; Shinohara et al., 2002; Kazahaya et al., 2002]. Moreover, different processes suggested in literature could explain fluctuations of the gas flux or composition measured either
in plumes, fumaroles or in thermal springs. A series of distinct degassing events may be generated by magma rise at shallow depth in discontinuous separate events [Nuccio and Paonita, 2001]. Enhanced by the heat dissipated from a close magma, processes of hydrofracturing [Rowe et al., 1992], thermal cracking or sealing [Edmonds et al., 2003] have been proposed to explain episodic release or trapping of magmatic fluids. During gas transfer to the surface, variations of the permeability of the hydrothermal system, often due to surficial sealing processes, may occur [Zlotnicki et al., 1992; Fischer et al., 1994; Harris and Maciejewski, 2000]. Thermochemical models show that decompression, cooling, oxidation processes, interactions with wall rocks and with the hydrothermal system, may lead to large variations in the volatile proportions in volcanic plumes [Giggenbach and Sheppard, 1989; Symonds et al., 2001]. Eventually, the proportion of volatile components in surface waters may be modified by the dynamics of the aquifers varying due to external influences, such as the seasonal supply of meteoric waters [e.g., Ingebritsen et al., 2001; Lo´pez et al., 2006]. Most of these mechanisms explain the observation of inter-mittent gas fluctuations but not their nonrandom repetition over a long time interval. This may be directly linked to a magmatic source that expels its fluids episodically.
[3] As an example, La Soufrie`re de Guadeloupe
under-went an important volcanic crisis in 1975 – 1977 with many phreatic explosions [Feuillard et al., 1983; Komorowski et al., 2005]. Since then, thermal springs have been sampled about twice a month and show a pattern of episodically
1Institut de Physique du Globe de Paris, E´ quipe de Ge´ologie des
Syste`mes Volcaniques, Paris, France.
2Now at Department of Geography, University of Cambridge,
Cambridge, UK.
3Universite´ Pierre et Marie Curie, Paris, France.
Copyright 2008 by the American Geophysical Union. 0148-0227/08/2007JB005130$09.00
changing hydrothermal fluid compositions (Figure 1a). Such a sustained monitoring data set is rare for a nonerupting volcano. The phreatic crisis, the subsequent evolution of the thermal spring compositions, and the accompanying shallow seismicity have all been interpreted as the consequences of a shallow magma intrusion [Villemant et al., 2005].
[4] Our aim here is to develop a general model that
describes the episodic degassing of a shallow intrusion of andesitic magma. If a volcano acts as a closed system, with no recharge or loss of magma by eruption, and exhibits an episodic degassing behavior, it may be the signature of a degassing stored magma. In this case, the interpretation of real geochemical data with the model would provide infor-mation on this subsurface magmatic system. When a magma body stalls in the crust, as it loses heat to the host rocks, it cools and crystallizes. This leads to the oversaturation of
dissolved volatiles, which then exsolve and pressurize the magma intrusion [Blake, 1984; Tait et al., 1989]. The generated overpressure may exceed a threshold that induces wall rock failure. The high viscosity of highly crystalline andesitic magmas inhibits dike propagation into the opened fractures [Marsh, 1981; Rubin, 1995a, 1995b] and instead decompression can occur through gas escape. Then, the intrusion returns to its original pressure and a new cycle of cooling-crystallization-degassing occurs leading to episodic degassing. Gases rise rapidly to the surface forming fumar-oles, plumes, or interacting with the surficial hydrothermal system. The model therefore considers three main processes: the mechanisms of magma cooling and crystallization, the magma pressurization induced by melt crystallization and the process of episodic gas expulsion. The influence of the initial conditions of the magma intrusion on the different Figure 1. Episodic degassing observed at (a) La Soufrie`re de Guadeloupe (Lesser Antilles, France)
through the temporal variations in the chlorine content sampled at Carbet E´ chelle thermal spring over a period of12 years (reproduced from Villemant et al. [2005]), (b) Galeras (Colombia) through the time variation in SO2flux (metric tons per day, m d1) remotely measured by COSPEC between the eruptions
of 16 July 1992 and 14 January 1993 (reproduced from Fischer et al. [1996]), (c) Campi Flegrei (Italy) through the temporal variations in the gas composition expressed as CO2/H2O (molar ratio) from 1981
regimes of magma degassing is discussed. This allows the characterization of the initial conditions that play a key role in the degassing and the physical mechanisms by which they exert this strong influence. Observations of fossil intrusions [Marsh, 2000; White and Herrington, 2000] and of persistent degassing activities [Francis et al., 1993; Allard, 1997] support the idea that magma frequently does not reach the surface. It may stall at shallow depth in the crust and perturb volcanic and hydrothermal activity, both on short timescales with phreatic or phreatomagmatic erup-tions, but also on decadal timescales by sustained fumarolic degassing. This model is consequently of particular relevance for volcano monitoring and hazard assessment purposes.
2. Model Description
[5] Our model describes the degassing of an andesitic
magma following its intrusion at shallow depth. It is based on the following assumptions. Magma is a typical water-rich silicic andesite consisting of a rhyolitic melt (with 3 to 6 wt % H2O) with a significant quantity of crystals. Magma
is stored at a depth where it is volatile saturated. It has already lost some gas exsolved during its ascent. The magma intrusion, assumed spherical, evolves as a closed system. Because of the thermal contrast with the surround-ings, the magma convects to transfer heat to the country rock where it is then assumed to be diffused by conduction. In the reservoir interior, only the process of thermal con-vection is considered; compositional concon-vection and crystal settling are neglected [Martin and Nokes, 1989]. Magma cooling leads to melt crystallization, water exsolution and magma overpressure. An elastic homogeneous enclosing medium is considered. When the overpressure reaches a specific threshold, assumed constant with time, wall rocks fracture and the excess gas escapes instantaneously from the intrusion. After the gas expulsion, pressure returns to lithostatic, the system is closed again and a new cooling-crystallization-degassing cycle occurs. A series of such cycles leads to episodic degassing process. The model describes the degassing until the magma no longer convects. This mechanism has a short timescale with respect to the time of complete magma cooling. The global model may be divided in three stages: (1) magma cooling and tion, (2) magma pressurization induced by melt crystalliza-tion, and (3) episodic gas expulsion. In the following sections, we develop the theoretical framework of the model and provide the expressions of the model outputs that may be compared with geochemical monitoring data. Finally, we estimate the realistic range of model input values. Figure 2 summarizes the physical processes involved in magma degassing, the observable model outputs, the main variables and the model inputs. All symbols, notations and indices used are defined in Tables 1 and 2. Model inputs, variables, and outputs are listed in Table 3. The set of fixed physical parameters is given in Table 4.
2.1. Magma Crystallization, Cooling, and Viscosity Evolution
[6] The processes of magma cooling and crystallization
are described via the spatially averaged magma temperature and mass of crystals. A simplified model (Figure 2, step 1) is built with the following assumptions. Magma undergoes
thermal convection. Heat transfers to the surroundings are approximated by the simple case of a magma which cools due to contact with a cold vertical semi-infinite flat plate held at constant temperature. Heat loss is balanced by the variation of the magma temperature and the latent heat of crystallization. The evolution of the mean magma tem-perature with time can be formulated as follows [Spera, 1980]: dT dt ¼ 3ath R23bth athg n bthKth1bthðT TcontactÞ1þbth 1þ L CpðTlTsÞ : ð1Þ
Both coefficients ath and bth depend on the type of heat
transfer. We assume that the andesitic magmas are highly crystalline and viscous such that convection is laminar and bth is 0.25 [Bejan, 1984]; ath also depends on magma
geometry and is0.7 [Churchill and Usagi, 1972]. [7] A simple dependency of crystallization on the
tem-perature decrease is assumed:
1 M dmc dt ¼ 1 Tl Ts ð Þ dT dt : ð2Þ
Relation (2) is classically used for Ts< T < Tl, with a crystal
mass equal to zero at the liquidus temperature. Andesites contain a fraction of inherited crystals and the crystal mass is consequently not necessarily zero at T = Tl. Thus, initial
magma temperature and crystallinity are independent conditions.
[8] The contact temperature is approximated by that
obtained with models which describe both magma cooling and host rock heating by conductive heat transfer [Carslaw and Jaeger, 1986], assuming no hydrothermal convection in the surroundings. For similar thermal diffusivity of magma and wall rocks, Tcontactmay be considered constant and, to a
first approximation, given by
Tcontact¼ 1 2 Tiþ Tsurroundings ¼1 2ð2Ti DTÞ: ð3Þ
Relation (3) is valid while t < 0.4 R2/Kth, a condition always
met in our study of the first stages of the cooling process. [9] According to equation (1), and because it may vary
significantly in crystallizing andesitic magmas, magma viscosity plays a key role in cooling. It is controlled by numerous variables, which may be dependent, including magma crystallinity, crystal sizes and shapes, magma com-position and temperature, and water content of melt and bubbles. When the crystal fractionF exceeds the percolation threshold of the solid particlesFPof the order of 40 vol %
[Rutgers, 1962; Shaw, 1969; Wildemuth and Williams, 1985; Ryerson et al., 1988; Lejeune and Richet, 1995], the viscosity may increase dramatically, by 5 – 6 orders of magnitude as F changes from 40 to 60% [Lejeune and Richet, 1995] (see Appendix A, Figure A1). Crystal content of silicic andesitic magmas is typically within the range 25 – 50 vol %, thus we assume that it is the primary control of magma viscosity.
[10] A rapid review of the role of the other variables is
Figure 2. Schematic description of the three steps of the model: (1) magma cooling and crystallization, (2) magma pressurization induced by melt crystallization, and (3) episodic gas expulsion. The physical processes taken into account and the involved model outputs, inputs and variables are detailed. The dependence relationships obtained (section 3) between the model outputs and relevant variables, and the model inputs are mentioned. The cyclic evolution is underlined with dashed lines. All used symbols are listed in Tables 1 and 2.
variations on viscosity. The percolation threshold lowers with decreasing solid particles sizes [Ward and Whitmore, 1950a, 1950b] and symmetry [Milliken et al., 1989]. How-ever, these effects are minor compared to the role of the crystal fraction [Lejeune and Richet, 1995]. The cooling of a rhyolitic melt by 200°C from an initial temperature of 950°C, typically produces an increase of the viscosity of 2 – 3 orders of magnitude [Neuville et al., 1993; Hess and Dingwell, 1996]. Melt crystallization also implies changes in melt composition manifested in an increase in silica content and consequently in melt polymerization and vis-cosity. Such well-known effects are of limited impact in this case. Isobaric cooling, crystallization and degassing lead to negligible variations in dissolved water contents (<0.1 wt %) and melt viscosity [Hess and Dingwell, 1996; Richet et al., 1996; Stevenson et al., 1998]. The presence of gas bubbles implies a viscosity decrease but in a limited range if bubble fraction is 40 vol % [Lejeune, 1994]. This is the case in our model where exsolved gas periodically escapes from the magma. The combined influence of these different variables
is poorly known. Since the temperature decrease and the melt polymerization both increase the viscosity, assuming a rheology only dependent on the crystal fraction underesti-mates the viscosity variations with time. The proposed analytical viscosity law and the method to determine it are given in Appendix A.
Table 1. Latin Symbols Useda
Symbol Definition
ath thermal transfer coefficient
A ratio of the perfect gas constant to the
molar mass of the volatile species, J kg1K1
Aincl inclination of the parallel asymptotes of the viscosity curve
bth thermal transfer coefficient
Basympt distance between the asymptotes of the viscosity curve
Cslope slope at the inflexion point of the viscosity curve
CP silicate melt heat capacity, J kg1K1
g acceleration due to gravity, m2s1
k cooling-crystallization-degassing cycle number
Kth silicate melt thermal diffusivity, m2s1
L magma crystallization latent heat, J kg1
m mass, kg
G total mass fraction of expelled gas with the initial
available mass of gas just after intrusion
M total mass of magma, kg
n exponent of the water solubility law written x = sPn
nR exponent of the Roscoe law
N gas pulse number since magma intrusion
P magma pressure, Pa
R magma intrusion radius, m
s water solubility coefficient
t time, s
~t dimensionless time
t rescaled time relative to the time until convection ceases
tconv time to cessation of convection, s
tC time to the transition in the cooling process, s
tD time to the transition in the degassing process, s
T average magma temperature, K
Tcontact contact temperature between magma and surroundings, K
Tsurroundings temperature of the surrounding rocks, K
Ti initial magma temperature just after intrusion, K
Tl magma liquidus temperature, K
Ts magma solidus temperature, K
x weight fraction of the volatile species dissolved in the
melt md/ml
z magma intrusion depth, m
a
Subscript k refers to the initial variables values for a given cycle k, and any variable topped by an overbar refers to the final variable value, just before fracturing. Subscript i refers to the initial variables values just after intrusion. Any variable topped by a tilde refers to the dimensionless variable.
Table 2. Greek Symbols Useda
Symbol Definiton
ath silicate melt thermal expansion, K1
bliq bulk modulus of silicate liquid, Pa
h magma dynamic viscosity, Pa s
h0 dynamic viscosity of the homogeneous magmatic liquid, Pa s
hinflex dynamic viscosity at the inflexion point of the viscosity
curve, Pa s
F crystal volume fraction, vol %
Fi initial volume fraction of crystals, vol %
Finflex crystal volume fraction at the inflexion point of the viscosity
curve, vol %
Fm crystal volume fraction preventing any liquid movement, vol %
FP volume fraction of crystals at their percolation threshold, vol %
lg fraction of exsolved gas mass at cycle onset with the total mass of
magma (mg/M)
m rigidity of the surrounding rocks, Pa
m 4/3 times the host rock rigidity, Pa
v magma kinematic viscosity, m2s1
r magma density, kg m3
rlitho lithostatic density, kg m3
st tensile strength of the surrounding rocks, Pa
t characteristic time of magma cooling, s
Dk(m) increase of mass during a given cycle k, kg
DP magma overpressure, Pa
Dt time interval between two consecutive gas pulses, s
DG fraction of expelled gas mass per pulse with the initial available
mass of gas just after intrusion
DT initial temperature contrast between magma and surroundings, K
a
Subscripts l, c, g, d refer to the liquid, crystal, and exsolved and dissolved gas phases, respectively. Subscript k refers to the initial variables values for a given cycle k, and any variable topped by an overbar refers to the final variable value, just before fracturing. Subscript i refers to the initial variables values just after intrusion. Any variable topped by a tilde refers to the dimensionless variable.
Table 3. Model Inputs, Relevant Variables, and Outputsa
Model Inputs Symbol
Range (Reference Value) Initial conditions of magma intrusion
Initial crystal volume fraction (vol %) Fi 30 – 50 (40)
Magma intrusion radius (m) R 5 – 500 (200)
Initial temperature contrast between magma and surroundings (K)
DT 200 – 800 (400)
Magma intrusion depth (m) z [2 – 10] (3) 103
Initial condition of host rocks
Tensile strength of the host rocks (Pa) st [0.01 – 10] (1) 106
Relevant model variables
Magma temperature T
Magma dynamic viscosity h
Magma crystal content mc/M
Magma overpressure DP
Model outputs
Time interval between two consecutive pulses
Dt Gas pulse number since magma
intrusion
N
Mass fraction of expelled gas per pulse DG
Total mass fraction of expelled gas since magma intrusion
G
a
[11] The magma cooling law (equation (1)) may be
written in a nondimensional form:
d ~T d~t¼ n Fð Þi n F ~TDTþ Ti ~Tþ1 2 1þbth ; ð4Þ
with ~T = (T Ti)/DT and ~t = t/t. The characteristic time t,
which depends on the initial conditions (R, DT, Fi), is
defined by t R; DT ; Fð iÞ ¼ 1þ L CpðTlTsÞ 3ath R23bth athg n Fð Þi bth K1bth th DTbth : ð5Þ
Equation (4) is numerically solved using a Runge-Kutta method, from t = 0 corresponding to magma intrusion until magma crystallinity reaches 60 vol %. Beyond this value, magma behaves as a solid [Lejeune and Richet, 1995]. We restrict the model to the convective cooling regime.
2.2. Magma Pressurization Induced by Melt Crystallization
[12] Magma is composed of three phases: crystal, liquid,
and gas. It is in thermodynamic equilibrium, and all system variables are assumed homogeneous. Tait et al. [1989] expressed the magma pressure as a function of its crystallization taking into account the effects of the initial pressure, via the solubility law of volatiles in the melt, the deformation of the surrounding rocks assumed elastic, the initial mass of exsolved gas and the crystallization-induced contraction of the magma. Here, we extend this model by imposing an initial crystal content. In addition, we con-sider a cyclic host rock fracturing and gas escape. A schematic illustration of the model is given in (Figure 2, step 2). For a given cycle k, this leads to the equation of state for magma
fkðP; TÞ ¼ DkðmcÞ M ð6Þ with fkðP; TÞ ¼ 1 Pk P n 1mck M þ P Pk 1 PPk ATrlsP n 1 mþ 1 bliq ! 1mck M þ PPk ATrcsP n 1 m mck M þ lgk sPn " # 1þ 1 sPn P AT 1 rc 1 rl ¼ N1þ N2 1þ 1 sPn P AT 1 rc 1 rl : ð7Þ Table 4. Fixed Model Physical Parameters
Model Physical
Parameters Symbol Value Source
Characterizing Magma Intrusion
Silicate melt thermal expansion (K1) ath 5 105
Inclination of the parallel asymptotes of the viscosity curve
Arot 2.4 see Appendix A
Bulk modulus of silicate liquids (Pa) bliq 10
10
Touloukian et al. [1981]
Distance between viscosity law asymptotes Basympt 1.5 see Appendix A
Slope at the inflexion point of the viscosity curve Cslope 80.0 see Appendix A
Silicate melt specific heat (J kg1K1) CP 1000 Richet and Bottinga [1986]
Crystal volume fraction at the inflexion point of the viscosity curve (vol %)
Finflex 0.5 see Appendix A
Silicate melt thermal diffusivity (m2s1) Kth 106 Bagdassarov et al. [1996]
Melt crystallization latent heat (J kg1) L 3.3 105
Nicholls and Stout [1982]
Solubility law exponent n 0.5 Burnham [1975]
Magma dynamic viscosity at the inflexion point of the viscosity curve (Pa s)
hinflex 10 11
see Appendix A
Crystal density (kg m3) rc 3100
Silicate liquid density (kg m3) rl 2400 Spera [2000]
Solubility law coefficient (Pa1/2) s 4.11 106 Burnham [1962]
Initial magma temperature just after intrusion (K) Ti 1173 Scaillet and Pichavant [2003]
Magma liquidus temperature (K) Tl 1223 Spera [2000]
Magma solidus temperature (K) Ts 1023 Spera [2000]
Independent of Magma Intrusion
Thermal transfer coefficient ath 0.67 Churchill and Usagi [1972]
Thermal transfer coefficient bth 0.25 Bejan [1984]
Crystal volume fraction preventing any liquid movement (vol %)
Fm 0.6 see Appendix A
Crystal percolation threshold (vol %) FP 0.4 see section 2.1
Rigidity of surrounding rocks (Pa) m 1010 Touloukian et al. [1981]
Roscoe law coefficient nR 2.5 see Appendix A
The denominator in equation (7) describes the magma contraction due to melt crystallization. N1is rewritten
N1 1 Pk P n mlk M ; ð8Þ
with mlk/M = (1 (mck/M)) for mass conservation, where
the mass of gas is neglected relative to the masses of liquid and crystals. It describes the contribution of gas exsolution which results from the combined influence of the mass of residual melt and of the initial pressure via the water solubility law. Assuming the ideal gas law rg= P/AT, the
second term is rewritten
N2¼ P Pk x N2lþ N2cþ N2g ¼P Pk x 1 mþ 1 bliq ! rg rl mlk M þ 1 m rg rc mck M þ 1 Pk mgk M " # : ð9Þ
N2l, N2c, and N2g underline the role of the mass of liquid,
crystal and gas, respectively, at a cycle onset. They describe both the influence of the elastic deformation of the surroundings (terms proportional to 1/m) and the compres-sibility of the different phases. Crystals are assumed to be incompressible, the melt compressibility is equal to 1/bliq
and the gas compressibility is approximated by 1/Pkas the
temperature decrease is low during a cycle.
[13] During a given cycle k, the evolution of magma
pressure with time is linearized:
P Pkþ dP dt tk t tk ð Þ: ð10Þ
Combining equations (6) and (7), we can write
dP dtð Þ ¼t 1 M dDmc dt @fkðP;TÞ @T h i P dT dt @fkðP;TÞ @P h i T : ð11Þ
Given the crystallization law (equation (2)), the magma pressurization rate becomes
dP dtð Þ ¼ t dT dt 1 TlTsþ @fkðP;TÞ @T h i P @fkðP;TÞ @P h i T : ð12Þ
2.3. Episodic Gas Expulsion
[14] Wall rock failure occurs when magma overpressure
exceeds a critical value, which depends both on the mech-anism of fracturing and on the system geometry. We assume that fractures are opened in tensile regime, as commonly observed in experiments of forced fluid injections [Cornet, 1992]. In this case, host rock fracturing occurs when magma overpressure is twice the tensile strength of the surrounding rocks, assumed constant with time, because the intrusion is spherical [Tait et al., 1989]. Instantaneously, gas escapes and magma pressure returns to the lithostatic pressure. The magma decompression leads to an additional exsolution of a small amount of gas represented by the mass of exsolved gas at the onset of a cycle. This process results in a feedback
on magma pressurization (Figure 2, step 3). In section 2.3.1, the expression of the amount of exsolved gas induced by decompression is determined. In section 2.3.2, the simpli-fied formulations of the state equation (equations (6) and (7)) and of the pressurization rate for andesitic magmas are developed.
2.3.1. Mass Fraction of Exsolved Gas at a Cycle Onset [15] Magma decompression leads to the exsolution of a
fraction of volatiles, which remains trapped in the intrusion. This process controls the term N2g (equation (9)), which
describes the influence of the mass fraction of exsolved gas lgkat the onset of a cycle k. Conservation of the mass of
volatile species at the kth fracturing event implies
mdk¼ mg kþ1ð Þþ md kþ1ð Þ; ð13Þ where, for a given cycle, m refers to values just before fracturing (see Tables 1 and 2) because all the exsolved gas escapes from the magma at fracturing. Using the solubility law, equation (13) may be rewritten
s Pð kþ 2stÞnmlk¼ mg kþ1ð Þþ sPnkml kþ1ð Þ: ð14Þ If the mass of volatiles is neglected, the total mass conservation can be written
mlk M ¼ ml kþ1ð Þ M ¼ 1 mc kþ1ð Þ M : ð15Þ
Therefore the mass fraction of exsolved gas at the onset of cycle k is lgk¼ s Pð kþ 2stÞ n sPn k 1mck M : ð16Þ
It depends on the amount of residual melt and the initial pressure but, more importantly, on the tensile strength of the surrounding rocks, which determines the maximum overpressure and thus the magnitude of decompression. 2.3.2. Pressurization of Andesitic Magmas
[16] Given the range of assumed magma depths (2 –
10 km) and tensile strengths of the surrounding rocks (0.01 – 10 MPa) (see section 2.5), pressure variations are low compared to lithostatic pressure and the different terms of the state equation of magma fk (equation (7)) may
consequently be simplified as follows:
N1 n mlk M DP Pk ; ð17Þ N2l N1 1 n 1 sPn k rg rl 1 bliq þ1 m ! Pk; ð18Þ N2c N1 1 n 1 sPn k rg rc Pk m mck mlk ; ð19Þ N2g N1 1 n 1þ 2st Pk n 1 2st Pk : ð20Þ
For the set of fixed physical parameters, listed in Table 4, the ratio N2l/N1varies within the range 0.01 – 0.14. Because
of the high water solubility and the low liquid compressi-bility, the term N2l is negligible compared to the term N1.
The ratio N2c/N1 is significant only when the mass of
residual melt is low. This is not the case in our study, which describes the first stages of crystallization. The value of N2g/N1 is not well constrained and varies from 104 to
101. Therefore, the term N2g cannot be neglected,
especially for high tensile strengths. Finally, the term related to the crystallization contraction varies between 0.3 and 0.7 and has to be accounted for. Neglecting the contribution from the host rock deformation and from the liquid compressibility, the magma equation of state fkmay
therefore be approximated by fkðP; TÞ 1 Pk P n 1mck M þlgk sPn P Pk 1 1þ 1 sPnATP r1 c 1 rl ; ð21Þ which gives @fk @P T Pk;Tk ð Þ ¼ n Pk 1 mck M þ lgk sPnþ1 k 1þ 1 sPn k Pk ATk 1 rc 1 rl ð22Þ and @fk @T P Pk;Tk ð Þ ¼ 0: ð23Þ
From the magma cooling law (equation (1)), the pressuriza-tion of andesitic magmas, induced by melt crystallizapressuriza-tion and water exsolution, may thus be described by
dP dt ð Þ ¼tk 3ath R23bth athg nk bth K1bth th ðTk TcontactÞ1þbth Tl Ts ð Þ þL Cp 2 6 4 3 7 5 n Pk 1 mck M þ lgk sPnþ1 k 1þ 1 sPn k Pk ATk 1 rc 1 rl 2 4 3 5 1 : ð24Þ 2.4. Model Outputs
[17] We choose model outputs that may be compared to
geophysical and geochemical data characterizing the surfi-cial manifestations of a degassing magma: the time interval Dt between two consecutive gas pulses, the number N of gas pulses since magma intrusion until a given time, the mass fraction DG of expelled gas per cycle relative to the initial available mass of gas, and the total mass fraction G of expelled gas. In practice, Dt is more easily estimated because it does not require a gas survey starting at the time of magma intrusion. These outputs are of particular interest because they are directly observable, contrary to other model variables such as magma viscosity, temperature, crystal content and overpressure. The relevant model vari-ables are listed in Table 3.
[18] Mass conservation for the volatile species during a
given cycle k yields the following expression of the mass fraction of exsolved gas at the end of the cycle:
mgk M ¼ lgkþ xk 1 mck M xk 1 mck M : ð25Þ
This equation may be rewritten
mgk M ¼ lgkþ sP n k 1 mck M s Pð kþ 2stÞ n 1mck M : ð26Þ
The entire exsolved gas is lost at each fracturing event and the cumulative fraction of expelled gas after N cycles is therefore given by mg M XN k¼1 lgkþ sPnk 1 mck M s Pð kþ 2stÞ n 1mck M : ð27Þ
It follows that G, the total mass of expelled gas relative to the initial available mass of gas (mg)Maxjust after intrusion,
is G¼ mg mg Max ¼XNk¼1DkG; ð28Þ
with DkG the mass fraction of gas expelled at a given
cycle k relative to the initial available mass of gas
DkG lgkþ sPkn 1 mck M s Pð kþ 2stÞn 1mMck sPn i 1 mci M : ð29Þ
2.5. Realistic Range of the Model Input Values [19] Model inputs are based on a set of initial conditions
appropriate for (1) the host rocks, with the tensile strength of the surrounding rocksst, and (2) the magma intrusion,
including: the intrusion radius R and depth z, the tempera-ture contrastDT between magma and surroundings, and the magma crystallinity Fi. They may be divided into two
classes. The first class is composed of R,DT and Fi, which
directly control the magma cooling and crystallization processes (Figure 2, step 1). The second class includes z andst, which are involved in the second and third stages of
the model, respectively, describing magma pressurization (Figure 2, step 2) and episodic gas expulsion (Figure 2, step 3). Realistic ranges of these model input values may be determined for andesitic magmas.
[20] The model has been run for a range of plausible
intrusion sizes (Table 3). Estimation of Fi values may be
based on petrological studies of andesitic magmas. Most microlites and nanolites form during rapid magma ascent, partly due to melt cooling but mainly to magma degassing. Phenocrysts are mainly formed within magma chamber during slow cooling processes. Thus, we use the phenocryst contents observed in andesitic magmas to estimate the range ofFivalues. This represents an upper limit of the intrusion
crystallinity because rapid phenocryst overgrowth (a few months or less) may happen during magma ascent [Blundy and Cashman, 2005]. Ranges are 25 – 50 vol % at Mount Pele´e (Martinique, Lesser Antilles) [Villemant and Boudon, 1999]; 35 – 45 vol % at Soufrie`re Hills Volcano (Montserrat, Lesser Antilles) [Sparks et al., 2000]; 38 – 49 vol % at Mount St. Helens (United States) [Kuntz et al., 1981]; and 23 – 28 vol % at Mount Unzen (Japan) [Nakada and Motomura, 1999].Fiis thus assumed to range between 25 and 50 vol %.
[21] The initial temperature contrast depends on the
magma depth and on local thermal anomalies, such as those related to hydrothermal systems, which are generally poorly known. The value of DT may vary to a large extent from case to case and a wide range of 200 – 800 K is assumed.
[22] Constraints given by phase equilibrium and melt
inclusion composition indicate that the saturation pressure conditions vary between 1.1 and 2.2 kbar [Scaillet and Pichavant, 2003]. This corresponds to depth between 4.5 and 9 km at lithostatic equilibrium. This range represents upper limits of the magma intrusion depth, which is thus restricted to 2 – 10 km in the model.
[23] Studies of dike sizes [Pollard, 1987] and propagation
[Einarsson and Brandsdottir, 1980; Rubin and Pollard, 1987] may be used to constrain magma overpressures within the range 1 – 10 MPa. On the basis of laboratory experiments, host rock tensile strengths of 10 MPa are estimated for granites and basalts [Touloukian et al., 1981]. Such ranges of st values may be considered as upper
estimations since, in natural volcanic systems, surrounding rock material may be weakened by fissures and faults. Consequently, we allow for a very wide interval of possible values for st: from 0.01 to 10 MPa. The chosen ranges of
values for each model input are summarized in Table 3.
3. Influence of the Initial Conditions on Magma
Degassing
[24] Here, we introduce the evolution of magma
temper-ature, viscosity, crystal content and of the model outputs describing magma degassing (Table 3). Different regimes of magma cooling and gas expulsion are distinguished, depending on the initial conditions. In our treatment, we vary the value of one of the model inputs, while holding the others at their midrange or reference value (Table 3). We note that the initial conditions associated to the magma
intrusion are unknown in our study and may be considered as parameters. Computations are carried out for a set of fixed physical parameters (Table 4). This discussion will allow us to determine the initial conditions that play a key role in the degassing process.
3.1. Regimes of Magma Cooling
3.1.1. Time Rescaling of the Magma Cooling Process [25] The magma cooling law (equation (4)) is an equation
with separate variables ~T and ~t. Integration shows that the dimensionless temperature depends only onDT and Fi:
~
T ¼ ~TDT ;Fið Þ:~t ð30Þ
In this notation, variables are in brackets and parameters shown as indices. We note that R has no influence on the evolution of ~T with ~t. This evolution is shown in Figures 3 and 4 withDT and Fi, respectively, chosen in their range of
Figure 3. Evolution of dimensionless magma temperature ~
T with dimensionless time ~t, for values of the initial temperature contrast between magma and surrounding rocks DT within the range [200 – 800] K (Fi= 0.4). End points of
the curves correspond to the time to convection cessation, defined by a magma crystallinity of 60 vol %.
Figure 4. Evolution of dimensionless magma temperature ~
T with dimensionless time ~t, for values of the initial crystal content Fi (a) lower than the percolation threshold FP,
within the range 25 – 40 vol %, with the limit case Fi =
45 vol % for comparison, and (b) greater thanFP, within the
range 45 – 50 vol %, with the limit caseFi= 40 vol % for
comparison.DT is fixed at its reference value of 400 K. End points of the curves correspond to the time to convection cessation, defined by a magma crystallinity of 60 vol %.
values (Table 3). For any value of DT and for a reference value of Fi chosen at midrange, the magma temperature
follows a similar evolution, characterized by two cooling rates (Figure 3). Consequently, the cooling process may be rescaled by the time to cessation of convection ~tconv, which
can be written
~tconvðDT ; FiÞ ¼
hð ÞFi
DTa1; ð31Þ
where h is a function depending onFianda1a constant. A
least squares fit of the computed values of ~tconv with DT
within the range [200 – 800] K gives a1 1.1 (correlation
coefficient of 0.99).
[26] The evolution of magma temperature, for a set of
values of the initial crystal content and a reference value of DT, is more complex because different cooling trends are observed depending onFivalues. For Fivalues within the
range 25 – 40 vol %, magma cooling is characterized by two different cooling rates (Figure 4a) as mentioned before. For Fivalues greater than the crystal percolation threshold FP,
no such transition is observed. In the extreme case Fi =
50 vol %, viscosity is almost constant and the temperature decrease is monotonous with time (Figure 4b). In what follows, we focus on the case where two regimes of magma cooling are observed. Similarly to the relation determined previously as a function of DT variations, the time until convection ceases follows a specific law depending onFi
andDT. Given equation (31), we can write
~tconvðFi;DTÞ ¼
C DTa1Fa2
i
; ð32Þ
where C anda2are two constants. A least squares fit of the
computed values of ~tconv with Fi values in the range 25 –
40 vol % gives C 1360 and a2 1 (correlation coefficient
of 0.99). ForFi FP, the evolution of temperature can be
expressed as a function of t: t ¼ ~t ~tconv ¼ t tconv ; ð33Þ
where tconvis the time to cessation of convection, which is
dependent on R,DT and Fiaccording to the expression
tconvðR;DT ; FiÞ ¼ C
t R; DT ; Fð iÞ
DTa1Fa2
i
: ð34Þ
In Table 5, we report the estimated range of the values of tconv
as a function of (R,DT, Fi) when these model inputs are set to
values within the range discussed in section 2.5. From these calculations, we find that tconvmainly depends on the magma
intrusion radius:
tconv tconvð Þ:R ð35Þ This implies the same property for the rescaled time:
t tRð Þ:t ð36Þ Magma temperature consequently becomes
~
T ¼ ~TDT ;Fið Þ:tR ð37Þ
3.1.2. Magma Cooling Transition due to Crystal Percolation
[27] Both the magma viscosity increase and the decrease
of the temperature contrast between magma and surround-ings lead to a slowing of the magma cooling and crystal-lization processes. For any values of the initial conditions (DT, Fi), the temperature decrease with the rescaled time t
is characterized by two different cooling rates (Figure 5a). During the first cycles after magma intrusion, the cooling rate is large and constant then it changes rapidly to a constant rate about twenty times lower. We find that the transition between these two regimes corresponds to the strong viscosity increase due to the crystal percolation (Figures 5a, 5b, and 5c). Magma viscosity is consequently the key variable controlling temperature. This transition occurs at a critical rescaled cooling time tC, which is
relatively independent of DT and Fi. According to
equa-tions (33) and (35), the critical time tC is given to a first
approximation by
tC tCtconvð Þ:R ð38Þ and is thus mainly dependent on the value of R, the values of DT and Fi having only second-order effects. We can
consequently write
tC tCð Þ;R ð39Þ where tC is an increasing function of R: the larger the
magma intrusion, the more delayed the transition of the cooling regime.
[28] As observed in Figure 5c, the evolution of the crystal
content with the rescaled time t, weakly depends onDT:
mc M mc M h i Fi tR ð Þ: ð40Þ
Magma viscosity (Figure 5b) is directly dependent on the crystal content and is controlled by the same model inputs:
h hFið Þ:tR ð41Þ
3.2. Regimes of Episodic Gas Expulsion
[29] The influence of each model input (DT, Fi, z,st) on
magma degassing is now analyzed. The evolution of the model outputs (Dt, N, G, DG) with the rescaled time t is evaluated as a function of the different inputs and summa-rized in Figure 6. Thus, we determine the initial conditions that play a key role in the degassing process.
Table 5. Range of the Values of the Time tconv to Convection
Cessation as a Function of (R,DT, Fi), According to Equation (34)a Cooling Model Inputs Realistic Range of Values Range of tconv(years) Ratio max(tconv)/min(tconv)
R [5 – 500] m [1 – 457] 3 102
DT [200 – 800] K [377 – 56] 7
Fi [30 – 50] vol % [176 – 145] 1
aCalculations are carried out by varying the value of one of the model
input in the realistic range discussed in section 2.5, while holding the others
at their midrange or reference value (Table 3), with C = 1360,a1= 1.1, and
a2= 1. The ratio of the maximum value of tconv to its minimum value
3.2.1. Number of Gas Pulses and Time Interval Between Two Consecutive Pulses
[30] The degassing process is episodic and the evolution
of the number N of gas pulses is a step function. For any values of the model inputs (DT, Fi, z, st), this evolution
follows a similar trend when plotted as a function of the rescaled time t (Figure 7). The gas pulse frequency, which is the slope of the curve of N with respect to t, is large for the first cycles then rapidly drops defining two distinct regimes. These two modes are, of course, also present in the evolution of the time intervalDt between two consecutive pulses (Figure 8). N and Dt present the same dependence relationships with the model inputs. In what follows, these relations are discussed using the evolution of the pulse number. However, the transition between the degassing regimes is described with the evolution of Dt, because it is more clearly visible. It occurs at a critical time tD(D for
degassing) associated with the sharp increase in the evolu-tion ofDt. tDis primarily dependent onFiand varies within
the range 0.01 – 0.05 according to Figure 9. It is very close to the transition time of the cooling process tC. The decrease
of the cooling and crystallization rates with time slows down the gas exsolution and the magma pressurization. The time needed to reach the critical pressure for failure increases, and the gas pulse frequency decreases with time because gas exsolution induced by crystallization is the main process controlling the magma pressurization rate. The timescale and trend of the magma degassing is thus gov-erned by this process.
[31] According to equations (33) and (35), the time to the
transition in the degassing process tDis given by, to a first
approximation,
tD tDtconvð Þ:R ð42Þ The variations of tDwithFiare negligible compared to the
variations of tconvwith R. Consequently, tDmainly depends
on R:
tD tDð Þ;R ð43Þ
Figure 6. Evolution of the model outputs as a function of the different inputs (DT, Fi, z,st), for a particular rescaled
time t. Arrows specify the sense of variation while the information in brackets indicates the amplitude of variation at the time of convection cessation.
Figure 5. Evolution of (a) dimensionless magma tem-perature ~T , (b) magma dynamic viscosityh, and (c) crystal mass content with rescaled time t for extreme values ofDT (Fi fixed at its reference value of 40 vol %) (line) andFi
(DT fixed at its reference value of 400 K) (dots), respectively, in their realistic range of values. The dashed line corresponds to the variable evolution for the reference values of the model inputs. The zone delimited by two vertical lines indicates the range of the time tC to the
where tD is an increasing function of R: the larger the
magma intrusion, the more delayed the transition between the degassing regimes.
[32] Now we discuss the magnitude of the degassing
process in terms of the number of gas pulses, according to each model input (Figure 7). It is mainly the tensile strength of the surrounding rocks that controls the magnitude of the degassing (Figure 7c). This model input particularly con-strains the critical overpressure needed for wall rock failure (see section 2.3). Given the proportionality of the critical overpressure withstand because, to a first approximation,
magma overpressure increases linearly with time during a cycle, a normalized pulse number ~N may be introduced with
N¼ ~Nst refð Þ st
; ð44Þ
withst (ref )a reference value of the host rock tensile strength
which is arbitrarily chosen. Similarly,
Dt¼ D~t st st refð Þ
: ð45Þ
At a given time, the greater st, the larger the critical
overpressure and the smaller the pulse number. The influence of the critical overpressure on N is significant because it varies over several orders of magnitude, according to the range of values ofst. Consequently, fracturing is the major
process controlling the magnitude of magma degassing. [33] For a given scaled time t, the pulse number increases
with the depth of magma storage z (Figure 7d). The magma depth influences the degassing via the gas solubility law, which governs the volatile exsolution. According to this Figure 7. Evolution of the normalized number ~N of gas pulses (equation (44) withst (ref )taken equal to
1 MPa) plotted as a function of the rescaled time t, for different values of (a)DT, the initial temperature contrast between magma and surrounding rocks in [200 – 800] K (black lines underline the transition between the different degassing regimes), (b) Fi, the initial magma crystallinity, (c) st, the tensile
strength of the host rocks, and (d) z, the magma depth, in their realistic range of values (Table 3). The other model inputs are fixed at their midrange or reference value (Table 3).
law, the deeper the magma intrusion, the larger the quantity of dissolved gas available for exsolution. Consequently, for a given increase of the crystal mass, the magma pressurization and the host rock failure are all the more rapid and the pulse number important that z is large. When convection ceases, the pulse number is proportional topffiffiz. This is in agreement with the solubility of water in a rhyolitic melt, which is propor-tional to the square root of the lithostatic pressure.
[34] At a given time t, the pulse number increases with
decreasing initial crystal content (Figure 7b). A lower Fi
induces a lower magma viscosity. It follows that the cooling and crystallization rates are consequently higher. Finally, this implies a larger pressurization rate and a higher pulse number at a given time.
[35] Figure 7a shows that variations of the initial
temper-ature contrastDT do not significantly modify the evolution of N with t, as expected from the weak role of this input in the crystallization process.
[36] Figure 7 also shows that the number of gas pulses is
quite independent ofDT and varies, in order of decreasing importance, with st (3 magnitude orders), z and Fi. In
conclusion, N is mainly dependent onst:
N Nstð Þ:tR ð46Þ
Equivalently,Dt is approximated by
Dt Dtstð Þ:tR ð47Þ
During the first cycles after magma intrusion, the degassing rate is high and approximately constant then it decreases to a new rate ten times lower. According to equation (44) and to the chosen value ofst(Table 3), between 2 and 2000 gas
pulses may be observed until convection ceases. According to Figure 8, the scaled pulse frequency D~t1 is equal to
100 days in the first regime. In the second, the scaled frequency is about ten times greater D~t2 1200 days
(Figure 8). Given equation (45), it follows thatDt1ranges
between 1 day and 3 years andDt2between 10 days and
30 years, depending on the value ofst.
3.2.2. Mass Fraction of Expelled Gas per Pulse [37] For any value of Fi,st, and z, the mass fraction of
expelled gas per pulse, relative to the initial available mass of gas just after intrusion, globally decreases with time (Figure 10). To a first approximation,DG is proportional to the critical pressure needed for host rock failure. It conse-quently increases withstand may be normalized by
D ~G¼ DGst refð Þ st
: ð48Þ
In a second-order approximation, two patterns are observed according to the value of the tensile strength of the host rocks. For st 0.5 MPa, D ~G presents an oscillation
occurring close to the critical transition time tDleading to
maximum relative variations of 50%. It is all the more significant that st is high. Forst< 0.5 MPa, this effect is
negligible andDG decreases with time.
[38] Figure 11 shows that the mass of expelled gas varies,
in order of decreasing importance, with st, z, and Fi,
similarly with the number of gas pulses:
DG DGstð Þ:tR ð49Þ
The largest mass of gas expelled per pulse escapes in the first cycles and represents between 0.02 and 20% of the initial mass of gas available in the magma intrusion, depending on the value ofst (Table 3).
3.2.3. Total Mass Fraction of Expelled Gas
[39] Figure 12 illustrates the evolution of the total mass
fraction G of expelled gas, relative to the initial available mass of
Figure 8. Evolution of the normalized time interval D~t between gas pulses (equation (45) withst (ref )taken equal to
1 MPa) plotted as a function of the rescaled time t, for different values of the tensile strength of the host rocksstin
[0.01 – 5] MPa. The other model inputs are fixed at their midrange or reference value (Table 3).
Figure 9. Evolution of the time intervalDt between two consecutive gas pulses, in days, plotted as a function of the rescaled time t, for different values of the initial crystal content Fi in [25 – 40] vol %. The zone delimited by two
vertical lines indicates the range of the time tDto transition
in the degassing process. It is defined as the start time of the sharp increase ofDt with time. The other model inputs are fixed at their midrange or reference value (Table 3).
gas just after intrusion. It has the same characteristic trend as the pulse number N, with a changing slope close to the transition time of the degassing process tD. However, it is
independent of the tensile strength of the surrounding rocks and magma depth. Indeed G depends both on the number of gas pulses and the mass of gas expelled per cycle. Because DG varies little with time and thus may be assumed constant, G is approximated by G NDG and presents the same trend as N with time. According to equations (44) and (48), G is consequently independent of st, to a first
approximation. The greater the tensile strength the longer the cycle duration, but also the larger the mass of gas expelled per pulse.
[40] For a given time, if we neglect the mass fraction of
exsolved gas lg at cycle onset, both the initial available
mass of gas (mg)Maxand the mass mgof expelled gas depend
on the square root of z according to the solubility law of water in the melt. Their ratio G is thus independent of magma depth.
[41] Therefore, to a first approximation, the total mass
fraction of expelled gas mainly depends on the initial crystal content:
G GFið Þ:tR ð50Þ
At the transition time, between 20 and 30% of the initial available mass of gas is expelled. The first pulses imply a high gas loss rate which drops to a constant rate approximately twenty times lower after this transition.
4. Conclusions
[42] For andesitic volcanoes, we interpret episodic
pat-terns of magma degassing, spanning a few years to decades, as the result of the cooling, crystallization and
Figure 11. Evolution of the mass fraction of expelled gas per pulseDG plotted as a function of the rescaled time t, for different values of (a) magma depth z in its realistic range of values and (b) initial magma crystallinity Fi in
[25 – 40] vol %. The other model inputs are fixed at their midrange or reference value (Table 3). The time tD to the
transition in the degassing process is specified. Figure 10. Different regimes observed in the evolution of
the normalized mass fraction of expelled gas per pulseD ~G (equation (48) withst (ref )taken equal to 1 MPa) plotted as a
function of the rescaled time t, according to the value of the tensile strength of the host rocksstin its realistic range. The
other model inputs are fixed at their midrange or reference value (Table 3). The time tDto the transition in the degassing
degassing of a magma intrusion at shallow depth. When the magma crystallinity exceeds the crystal percolation threshold, magma viscosity significantly increases. This process strongly controls the time evolution of all the variables involved in the model (Figure 13). Two regimes of cooling are predicted with time, characterized by different cooling rates. They imply two modes of degass-ing defined by a high frequency of gas pulses at the beginning of the process and then by a frequency about ten times lower. This change in the degassing regime is the main characteristic of the proposed model.
[43] The trend and timescale of the degassing process are
controlled by the gas exsolution induced by melt crystalli-zation. The transition between the degassing regimes occurs
at a specific time dependent on the stored magma volume to a first approximation. Between 20 and 30% of the initial available mass of gas has already been expelled by this time. At a given time, the number of gas pulses, the time interval between two consecutive pulses, and the mass fraction of gas expelled per pulse are governed by the mechanism of host rock fracturing, which depends on the wall rock tensile strength.
[44] The comparison between model outputs and gas
data allows estimation of the key parameters associated with the degassing magmatic system. Even though the gas transfer from the magma intrusion up to the surface may be modified by a hydrothermal system, we expect that the tensile strength of the surrounding rocks may be deter-Figure 12. Evolution of the total mass fraction G of expelled gas plotted as a function of the rescaled
time t, for different values of (a) initial magma crystallinityFi, (b) tensile strength of the host rocksst,
and (c) magma depth z, in their realistic range of values (Table 3). The other model inputs are fixed at their midrange or reference value (Table 3). The transition time of the degassing process tDis specified.
Figure 13. Evolution with rescaled time t for the model inputs at their reference values (DT = 400 K, Fi= 0.4,st= 1 MPa
and z = 3 km) of (left) the relevant model variables (magma dynamic viscosityh, crystal mass fraction mc/M, dimensionless
temperature ~T and overpressure DP); (right) the normalized model outputs describing gas expulsion (D~t, ~N , G, D ~G). Variables with a tilde refer to the normalized values of the variables according to equations (45), (44), and (48) withst(ref)
mined by geochemical survey with suitable sampling frequency. After magma intrusion, the time interval be-tween pulses predicted by the model ranges bebe-tween 1 d and 3 years, depending on the value ofstwithin the range
0.01 – 10 MPa. The mass of expelled gas per pulse is almost constant with time, within the range 0.01 – 10% of the initial available mass of gas. Using a sustained monitoring record, the identification of a transition in the magma degassing regime would provide an estimate of the stored magma volume. Assuming a realistic range of R values to be 5 – 500 m, i.e., corresponding to a magma volume of 0.5 106 to 0.5 km3, this transition is predicted to occur within the range 3 weeks to 16 years after the magma emplacement.
[45] The estimation of the size of a magma intrusion is of
particular relevance for monitoring purposes. An intrusion may perturb the volcanic system over a long duration by degassing. Moreover it may considerably weaken the vol-canic edifice and favor flank collapses, as often observed in andesitic volcanoes. A future paper (M. Boichu et al., Degassing at La Soufrie`re de Guadeloupe volcano (Lesser Antilles) since the last eruptive crisis in 1975 – 1977: Result of a shallow magma intrusion?, unpublished manuscript, 2007) applies the model to interpret the long-term geochem-ical data collected at La Soufrie`re de Guadeloupe volcano (Lesser Antilles), which exhibits two distinct regimes of magma degassing, illustrating the potential relevance of this model for volcanic hazard assessment.
Appendix A: Analytical Dependence Law of
Viscosity on Crystal Fraction
[46] We propose an analytical law for the dependence of
magma viscosity on crystal volume fraction, which
repro-duces the trend showed by the experimental results of Lejeune and Richet [1995]:
logh¼ log hinflexþ AinclðF FinflexÞ
þ BasymptArctan C slopeðF FinflexÞ: ðA1Þ This relation depends on five nonindependent parameters. Finflex andhinflexrepresent the crystal volume fraction and
the magma viscosity at the inflexion point of the viscosity curve, Aincldefines the inclination of its parallel asymptotes,
Basymptis the distance between them, and Cslopeis the curve
slope at the inflexion point.
[47] The coefficient Ainclis constrained by the Roscoe law
h = h0ð1 F=FmÞnR [Roscoe, 1952], which describes the
linear variations of viscosity for a crystal fraction smaller than the percolation threshold; nRis an adjustable parameter
and Fm the crystal volume fraction preventing any liquid
movement. Most experimental measurements are well fitted for nR = 2.5 and Fm = 0.6 [Marsh, 1981; Lejeune and
Richet, 1995]. Viscosity of homogeneous rhyolitic liquids lies within the range 107– 109Pa s at 1200 K, according to theoretical and experimental studies [Hess and Dingwell, 1996; Neuville et al., 1993]. For a crystal fraction Fm,
magma behaves as a solid and viscosity reaches a relatively constant level around 1013– 1014Pa s, regardless of the type of material [Lejeune and Richet, 1995]. This controls the value of Basympt. Given h0, Aincl and Basympt, the value of
hinflex is constrained. Finally, Finflex and Cslope are both
linked to the percolation threshold associated to a crystal fraction of 40 vol %. Figure A1 illustrates the viscosity law that we propose for andesitic magmas.
[48] Acknowledgments. We are particularly grateful to Claude
Jaupart for many discussions about the model and to Clive Oppenheimer for his careful reading which helped to improve the manuscript. The manuscript benefited from the constructive and critical comments of David Pyle and an anonymous reviewer. This manuscript is the IPGP contribution 2340.
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![Figure 9. Evolution of the time interval Dt between two consecutive gas pulses, in days, plotted as a function of the rescaled time t, for different values of the initial crystal content F i in [25 – 40] vol %](https://thumb-eu.123doks.com/thumbv2/123doknet/14792664.602100/14.918.90.452.82.356/figure-evolution-interval-consecutive-plotted-function-rescaled-different.webp)
